On the isomorphism classes of weighted spaces of harmonic and holomorphic functions

Wolfgang Lusky¹

¹ Fakultät für Elektrotechnik, Informatik und Mathematik Universität Paderborn Warburger Straße 100 D-33098 Paderborn, Germany

Studia Mathematica, Tome 175 (2006) no. 1, pp. 19-45

Résumé

Let $ {\mit\Omega}$ be either the complex plane or the open unit disc. We completely determine the isomorphism classes of \[ Hv = \{ f: {\mit\Omega} \rightarrow \mathbb C \mbox{holomorphic}: \sup_{z \in {\mit\Omega}} |f(z)|v(z) \infty \} \] and investigate some isomorphism classes of \[ hv = \{ f: {\mit\Omega} \rightarrow \mathbb C \mbox{ harmonic} : \sup_{z \in {\mit\Omega}} |f(z)|v(z) \infty \} \] where $v$ is a given radial weight function. Our main results show that, without any further condition on $v$, there are only two possibilities for $Hv$, namely either $Hv \sim l_{ \infty}$ or $ Hv \sim H_{ \infty}$, and at least two possibilities for $hv$, again $hv \sim l_{ \infty}$ and $hv \sim H_{ \infty}$. We also discuss many new examples of weights.

Détail

DOI : 10.4064/sm175-1-2

Keywords: mit omega either complex plane unit disc completely determine isomorphism classes mit omega rightarrow mathbb mbox holomorphic sup mit omega infty investigate isomorphism classes mit omega rightarrow mathbb mbox harmonic sup mit omega infty where given radial weight function main results without further condition there only possibilities namely either sim infty sim infty least possibilities again sim infty sim infty discuss many examples weights